Can any sort of prosodic information (e.g. rhythm, intensity, pitch) be seen/understood just from a waveform alone (without reference to e.g. spectrograms)?
Definitely. Here's a waveform of my rendition of the word prosody with a declarative intonation:
Duration - As you can see, it's quite clear where the chunks of unobstructed periodic voicing as opposed to aperiodic noise are (though sometimes the exact placement of a boundary between the two can be tricky), and it's quite simple to measure their duration. In this case, the first and third vowels are comparable in length, while the middle vowel is much shorter. This makes sense, since the first vowel is stressed and the last vowel is unstressed but word-, phrase-, and utterance-final. The middle vowel is unstressed and word-medial.
What you might not be able to do is measure the respective durations of two adjacent sounds with similar acoustic properties--two fricatives, for example, or two vowels. In such cases we usually depend on being able to see the spectrum (to see where in the spectrum the frication noise is concentrated, or where formants are).
Intensity - The relative intensity of a sound can be gleaned from the amplitude of the waveform. It's clear that the first vowel is the most intense, while the second and third are quite similar to one another in their intensity.
Fundamental frequency ("pitch") - The average fundamental frequency (F0) of any portion of a voiced sound can be calculated by dividing the number of glottal pulses by the duration of the chunk in question. For example, if we zoom in on the first vowel...
...we can see that there are 16 glottal pulses in 0.152 seconds. 16 divided by 0.152 is about 105, so the average F0 of the vowel is about 105 Hz. If we do the same for the second and third vowels, we get 91 Hz and 86 Hz, respectively. So, as expected for declarative intonation, the stressed vowel, which is also the first vowel in the word, has the highest average F0, and the F0 drops for the second vowel and then drops a bit more for the last vowel.
Now, of course the F0 of a vowel does not usually stay exactly the same through the course of the vowel; if we wanted a more precise measurement we could measure the time between every two pulses and then divide 1 by that time to get the fundamental frequency for that part of the vowel. If we then plotted the results on a line graph we'd get a nice fundamental frequency contour.